Тема №1: Дискретные структуры, примеры



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1st theme

The 1st theme: Discrete structures, examples

Teacher:

Dotsent, PhD Sabirov Karimjon Kabildjanovich

Introduction

Mathematics

Discrete structures

  • Discrete structures are the study of processes occurring in nature or in some objects based on their state at some discrete moments of time or their discrete points
  • Example: the set of natural numbers

    the number 3 secede with neighbors 2 and 4

    Example for non-discrete is the interval

  •  

Mathematical structures

Mathematical structures can be classified into three main types:

  • algebraic structures
  • structures of orders
  • topological structures

Algebraic structures

If a rule is given for any two elements of the set, for uniquely determining the third element, then this is called the composition law.

A structure defined on the basis of the composition law is called an algebraic structure.

Axiomatic introduction of the operation:

  •  

Examples on algebraic structures

  • The set of natural numbers relative to the operations of addition and multiplication
  • The set of vectors in the plane relative to the addition operation
  • The set of matrices relative to the addition operation
  • Boolean structures (logical operation structures)

Structures of orders

  • If a relation 𝜌 is given between elements of some set X as follows
  • then it is said that the order structures are defined in the set X.

  •  

Examples on order structures

Topological structures

If on the set X the operation "[]" is given in the following way, then the topological structures are called:

  •  

Thank you for attentions!


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